The estimates on causality violating part of the wave function are given. The problem of acausal behavior of charge and current densities is discussed.
In relativistic theories the principle of microscopic causality states that ``information cannot travel faster than the speed of light'' \cite{kaku}. In the present work we show that the time evolution of relativistic wave functions violates this principle. We consider here the wave functions of massless and massive particles. In the case of massless particles the wave functions which violate the microscopic causality have an analytic form while in the case of massive particles we have to rely on numerical calculations. In both cases the wave functions which are strictly localized at t=0, at later times do not vanish {\it outside} the future light cone. \end{abstract}
We show that states of systems which, in a very general sense, are approximately localized at time t=0 in a finite region, with exponentially bounded tails outside, violate Einstein causality at later times. Implications are discussed.
The localization problem in relativistic quantum theory has persisted for more than seven decades, yet it is largely unknown and continues to perplex even those well-versed in the subject. At the heart of this problem lies a fundamental conflict between localizability and relativistic causality, which can also be construed as part of the broader dichotomy between measurement and unitary dynamics. This article provides a historical review of the localization problem in one-particle relativistic quantum mechanics, clarifying some persistent misconceptions in the literature, and underscoring the antinomy between causal dynamics and localized observables.